Direct and Inverse Solutions for Thermal- and Stress-Transients and the Analytical Determination of Boundary Conditions Using Remote Temperature or Strain Data

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
A. E. Segall ◽  
C. Drapaca ◽  
D. Engels ◽  
T. Zhu ◽  
H. Yang
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Thermophysical Properties ◽  
A Priori ◽  
Analytical Determination ◽  
Time Interval ◽  
Unknown Boundary ◽  
Direct Solution ◽  
Inverse Solutions

From an analytical standpoint, a majority of calculations use known boundary conditions (temperature or flux) and the so-called direct route to determine internal temperatures, strains, and/or stresses. For such problems where the thermal boundary condition is known a priori, the analytical procedure and solutions are tractable for the linear case where the thermophysical properties are independent of temperature. On the other hand, the inverse route where the boundary conditions must be determined from remotely determined temperature and/or flux data is much more difficult mathematically, as well as inherently sensitive to data errors (i.e., ill-posed). When solutions are available, they are often restricted to a harsh, albeit unrealistic step change in temperature or flux and/or are only valid for relatively short time frames before temperature changes occur at the far boundary. While the two approaches may seem to be at odds with each other, a generalized direct solution based on polynomial temperature or strain-histories can also be used to determine unknown boundary conditions via least-squares determination of coefficients. Once the inverse problem (and unknown boundary condition) is solved via these coefficients, the resulting polynomial can then be used with the generalized direct solution to determine the thermal- and stress-states as a function of time and position. When used for both thick slabs and tubes, excellent agreement was seen for various test cases. In fact, the derived solutions appear to be well suited for many thermal scenarios, provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties that do not vary with temperature. Since temperature dependent properties can certainly be an issue that affects accuracy in these types of calculations, some recent analytical procedures for both direct and inverse solutions are also discussed.

Transient Surface Strains and the Deconvolution of Thermoelastic States and Boundary Conditions

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
A. E. Segall ◽  
D. Engels ◽  
A. Hirsh
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
A Priori ◽  
Generalized Solution ◽  
Time Interval ◽  
Thermal Excitation ◽  
Unknown Boundary ◽  
Stress States ◽  
Natural Boundaries

Thermoelastic states as they pertain to thermal-shock are difficult to determine since the underlying boundary conditions must be known or measured. For direct problems where the boundary conditions such as temperature or flux, are known a priori, the procedure is mathematically tractable with many analytical solutions available. Although this is more practical from a measurement standpoint, the inverse problem where the boundary conditions must be determined from remotely determined temperature and/or flux data are ill-posed and therefore inherently sensitive to errors in the data. Moreover, the limited number of analytical solutions to the inverse problem rely on assumptions that usually restrict them to timeframes before the thermal wave reaches the natural boundaries of the structure. Fortunately, a generalized solution based on strain-histories can be used instead to determine the underlying thermal excitation via a least-squares determination of coefficients for generalized equations for strain. Once the inverse problem is solved and the unknown boundary condition on the opposing surface is determined, the resulting polynomial can then be used with the generalized direct solution to determine the thermal- and stress-states as a function of time and position. For the two geometries explored, namely a thick-walled cylinder under an internal transient with external convection and a slab with one adiabatic surface, excellent agreement was seen with various test cases. The derived solutions appear to be well suited for many thermal scenarios provided that the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties that do not vary with temperature. While polynomials were employed for the current analysis, transcendental functions and/or combinations with polynomials can also be used.

Thermoelastic Stresses in Thick-Walled Vessels Under Thermal Transients via the Inverse Route

2006 ◽  
Vol 128 (4) ◽  
pp. 599-604 ◽  
Author(s):  
A. E. Segall
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Thermal Stresses ◽  
Internal Surface ◽  
Time Interval ◽  
Unknown Boundary ◽  
Direct Solution ◽  
Thermal Transients ◽  
Thermoelastic Stresses ◽  
Walled Cylinder

A common threat to thick-walled vessels and pipes is thermal shock from operational steady state or transient thermoelastic stresses. As such, boundary conditions must be known or determined in order to reveal the underlying thermal state. For direct problems where all boundary conditions (temperature or flux) are known, the procedure is relatively straightforward and mathematically tractable as shown by many studies. Although more practical from a measurement standpoint, the inverse problem where the boundary conditions must be determined from remotely determined temperature and/or flux data is ill-posed and inherently sensitive to errors in the data. As a result, the inverse route is rarely used to determine thermal stresses. Moreover, most analytical solutions to the inverse problem rely on a host of assumptions that usually restrict their utility to time frames before the thermal wave reaches the natural boundaries of the structure. To help offset these limitations and at the same time solve for the useful case of a thick-walled cylinder exposed to thermal loading on the internal surface, the inverse problem was solved using a least-squares determination of polynomial coefficients based on a generalized direct solution to the heat equation. Once the inverse problem was solved in this fashion and the unknown boundary condition on the internal surface determined, the resulting polynomial was used with the generalized direct solution to determine the internal temperature and stress distributions as a function of time and radial position. For a thick-walled cylinder under an internal transient with external convection, excellent agreement was seen with known temperature histories. Given the versatility of the polynomial solutions advocated, the method appears well suited for many thermal scenarios provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties that do not vary with temperature.

Thermoelastic Stresses in Thick-Walled Vessels Under an Arbitrary Thermal Transient via the Inverse Route

2005 ◽  
Author(s):  
A. E. Segall
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Thermal Stresses ◽  
Thermal State ◽  
Time Interval ◽  
Unknown Boundary ◽  
Measured Temperature ◽  
Direct Solution ◽  
Thermoelastic Stresses ◽  
Walled Cylinder

A common threat to thick-walled vessels and pipes is thermal shock from operational steady-state or transient thermoelastic stresses. As such, boundary conditions must be known or determined in order to reveal the underlying thermal state. For direct problems where all boundary conditions are known, the procedure is relatively straightforward and mathematically tractable as shown by recent studies. Although more practical from a measurement standpoint, the inverse problem where boundary conditions must be determined from remotely determined temperature and/or flux data is ill-posed and inherently sensitive to errors in the data. As a result, the inverse route is rarely used to determine thermal-stresses. Moreover, most analytical solutions to the inverse problem rely on a host of assumptions that usually restrict their utility to timeframes before the thermal wave reaches the natural boundaries of the structure. To help offset these limitations and at the same time solve for the useful case of a thick-walled cylinder exposed to thermal loading on the ID, the inverse problem was solved using a least-squares determination of polynomial coefficients based on a generalized direct-solution to the Heat Equation. Once the inverse problem was solved in this fashion and the unknown boundary-condition on the ID determined, the resulting polynomial was used with the generalized direct solution to estimate the internal temperature and stress distributions as a function of time and radial position. For a thick-walled cylinder under an internal transient with external convection, excellent agreement was seen with various measured temperature histories. Given the versatility of the polynomial solutions advocated, the method appears well suited for many thermal scenarios provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties that do not vary with temperature.

Thoughts on the Deconvolution of Thermal- and Stress-States From Transient Histories

2008 ◽  
Author(s):  
A. E. Segall ◽  
D. Engels ◽  
A. Hirsh
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Generalized Solutions ◽  
Time Interval ◽  
Thermal Excitation ◽  
Unknown Boundary ◽  
Time Lags ◽  
Measured Temperature ◽  
Direct Solution ◽  
Stress States

When thermal-shock is an issue, the underlying thermal- and stress-states are often difficult to determine because the boundary conditions must be known. For direct problems where the boundary conditions such as temperature or flux are known a priori, the procedure is usually mathematically tractable and can therefore be solved analytically. On the other hand, the inverse problem where the boundary conditions must be determined remotely is inherently ill-posed and therefore sensitive to errors. Moreover, there are only limited numbers of analytical solutions and they are usually restricted to timeframes before the thermal wave reaches the natural boundaries of the structure. Fortunately, generalized solutions based on either measured temperature- and/or strain-histories can be used to determine the underlying thermal excitation via a least-squares determination of coefficients that require the direct solution to enforce the data. Once the inverse problem is solved and the unknown boundary-condition determined, the resulting polynomial can then be used with the generalized Direct solution to determine the thermal- and stress-states as a function of time and position. For the two geometries explored (thick-walled cylinder under an internal transient with external convection and a slab with one adiabatic surface), excellent agreement was seen with various test cases for histories based on either temperature or strain. However, the strain-based solutions may be preferable since they do not suffer from time-lags associated with thermally thick structures. The derived solutions appear to be well suited for many thermal scenarios provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties are independent of temperature.

scholarly journals Stiffness Estimation and Equivalence of Boundary Conditions in FEM Models

2021 ◽  
Vol 11 (4) ◽  
pp. 1482
Author(s):  
Róbert Huňady ◽  
Pavol Lengvarský ◽  
Peter Pavelka ◽  
Adam Kaľavský ◽  
Jakub Mlotek
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Boundary Conditions ◽  
Model Updating ◽  
Element Model ◽  
Computational Time ◽  
Stiffness Coefficients ◽  
First Case ◽  
Numerical Approaches

The paper deals with methods of equivalence of boundary conditions in finite element models that are based on finite element model updating technique. The proposed methods are based on the determination of the stiffness parameters in the section plate or region, where the boundary condition or the removed part of the model is replaced by the bushing connector. Two methods for determining its elastic properties are described. In the first case, the stiffness coefficients are determined by a series of static finite element analyses that are used to obtain the response of the removed part to the six basic types of loads. The second method is a combination of experimental and numerical approaches. The natural frequencies obtained by the measurement are used in finite element (FE) optimization, in which the response of the model is tuned by changing the stiffness coefficients of the bushing. Both methods provide a good estimate of the stiffness at the region where the model is replaced by an equivalent boundary condition. This increases the accuracy of the numerical model and also saves computational time and capacity due to element reduction.

Full Sliding Adhesive-Like Contact of V-Belts

2002 ◽  
Vol 124 (4) ◽  
pp. 706-712 ◽  
Author(s):  
Go¨ran Gerbert ◽  
Francesco Sorge
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Power Transmission ◽  
A Priori ◽  
Belt Drive ◽  
Natural Boundary ◽  
Condition Problem ◽  
New Type ◽  
Wrap Angle ◽  
Natural Boundary Conditions

Analysis of power transmission in a belt drive consisting of, e.g., two pulleys might be treated as a boundary value problem. Tight side tension FT, slack side tension FS and the wrap angle α are the three natural boundary conditions. In the literature, theories are developed where seating and unseating as well as the power transmitting part of the contact are considered. The solutions presented so far don’t fulfill the boundary conditions properly, since a certain tension ratio FT/FS is associated with a certain contact angle and not an a priori specified one. It appears that a new type of full sliding solution must be introduced to handle the boundary condition problem. During part of the contact there is almost no tension variation in spite of the full sliding conditions. The designation adhesive-like solution is here introduced for that part. Conditions and character of the adhesive-like solution are outlined in the paper.

scholarly journals An inverse spectral problem for Sturm-Liouville-type integro-differential operators with robin boundary conditions

2019 ◽  
Vol 50 (3) ◽  
pp. 207-221 ◽  
Author(s):  
Sergey Buterin
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Boundary Conditions ◽  
Differential Operators ◽  
A Priori ◽  
Sufficient Conditions ◽  
Robin Boundary Conditions ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Robin Boundary

The perturbation of the Sturm--Liouville differential operator on a finite interval with Robin boundary conditions by a convolution operator is considered. The inverse problem of recovering the convolution term along with one boundary condition from the spectrum is studied, provided that the Sturm--Liouville potential as well as the other boundary condition are known a priori. The uniqueness of solution for this inverse problem is established along with necessary and sufficient conditions for its solvability. The proof is constructive and gives an algorithm for solving the inverse problem.

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